Some Results on the Navier–Stokes Equations in Thin 3D Domains

Abstract

We consider the Navier–Stokes equations on thin 3D domains Qε=Ω×(0, ε), supplemented mainly with purely periodic boundary conditions or with periodic boundary conditions in the thin direction and homogeneous Dirichlet conditions on the lateral boundary. We prove global existence and uniqueness of solutions for initial data and forcing terms, which are larger and less regular than in previous works on thin domains. An important tool in the proofs are some Sobolev embeddings into anisotropic Lp-type spaces. Better results are proved in the purely periodic case, where the conservation of enstrophy property is used. For example, when the forcing term vanishes, we prove global existence and uniqueness of solutions if ‖(I−M)u0‖H1/2(Qε)exp(C−1ε−1/s‖Mu0 ‖2/sL2(Qε))⩽C for both boundary conditions or ‖Mu0‖H1(Qε)⩽Cε−β, ‖(Mu0)3‖L2(Qε)⩽Cεβ, ‖(I−M)u0‖H1/2(Qε)⩽Cε1/4−β/2 for purely periodic boundary conditions, where 1/2<s<1 and 0⩽β⩽1/2 are arbitrary, C is a prescribed positive constant independent of ε, and M denotes the average operator in the thin direction. We also give a new uniqueness criterium for weak Leray solutions.